Here \(e_2=(0,1)^T\) , \(v\)  is the velocity vector, \(p\)  is the pressure, \(\theta \)  is a scalar function.
The Boussinesq equations arises from many geophysical flows, such as atmospheric fronts and ocean circulations (see, for example, [1]},[2]}).
To understand the turbulence phenomena in fluid mechanics, one needs to go beyond classical solutions. The pair \((v, p, \theta )\)  on \(R^2\times R\)  is called a weak solution of (REF ) if they belong to \(L^2_{loc}(R^2\times R)\)  and solve (REF ) in the following sense:
\(\int _R\int _{R^2}(\partial _t\varphi \cdot v+v\otimes v: \nabla \varphi +p\hbox{div}\varphi +\theta e_2\cdot \varphi )dxdt=0,\nonumber \) 
